In the Euclidean Traveling Salesman Problem (TSP), we are given $n$ points and are asked to determine the shortest path that visits all of the points. Last year Sanjeev Arora astounded the algorithms community by announcing a polynomial-time approximation algorithm for Euclidean TSP. This problem has been long known to be NP-complete, and for nearly 20 years the best approximation known (due to Christofides) achieved an approximation factor of 1.5. Arora's most recent result is that, for any $c > 1$, it is possible to construct a tour that is within a factor of $(1+1/c)$ of optimal in time that is $O(n)$ times a polylogarithmic factor that depends on $c$ and the dimension of the space. In the plane the running time is $O(n (\log n)^{O(c)})$. This dramatic discovery combines two relatively simple devices: quadtrees and dynamic programming. In this talk I will present an informal and intuitive explanation of Arora's algorithm. I hope that the talk will be accessible to anyone with a basic knowledge of quadtrees and dynamic programming.